In this paper, we highlight possibilities for designing intangible cultural values into interactions with technologies in heritage spaces. We do this specifically through the design of ʔeləw ̓ k ̓ ʷ -Belongings, an interactive tangible table installed in a cultural heritage museum. The tabletop was collaboratively designed to communicate complex and narrative information and values about Musqueam culture. Rather than focusing only on content and interface design, we wanted visitors to also experience Musqueam values through their interactions with the system. We describe our value-sensitive design process, present five interdependent design goals, discuss the design strategies that enabled us to meet these goals, and evaluate our approach through a user study. From our design process and evaluation we offer recommendations for designing values into interactions more generally and for tangible interactions specifically in ways that support visitors' experience and understanding of specific cultural values through technology.